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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Let <span class="process-math">\({\bf x}^{(1)}(t), {\bf x}^{(2)}(t),\cdots, {\bf x}^{(n)}(t)\)</span> to be <span class="process-math">\(n\)</span> solutions of <span class="process-math">\((\ref{eq7_5})\text{.}\)</span> Then the necessary and sufficient conditions for <span class="process-math">\({\bf x}^{(1)}, {\bf x}^{(2)}, \cdots, {\bf x}^{(n)}\)</span> to be linear independent is for the matrix</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf X}=
\left(
\begin{array}{cccc}
x_1^{(1)}(t), &amp; x_1^{(2)}(t), &amp;\cdots, &amp; x_1^{(n)}(t)\\
x_2^{(1)}(t), &amp; x_2^{(2)}(t), &amp;\cdots, &amp; x_2^{(n)}(t)\\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots\\
x_n^{(1)}(t), &amp; x_n^{(2)}(t), &amp;\cdots, &amp; x_n^{(n)}(t)\\
\end{array}
\right),
\end{equation*}
</div>
<p class="continuation">we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\det {\bf X} \neq 0.
\end{equation*}
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<span class="incontext"><a href="sec6_1.html#p-253" class="internal">in-context</a></span>
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